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J. Phys. Chem. C 2007, 111, 1342-1346
Effect of Long-Range Attraction on Growth Model Tian Hui Zhang and Xiang Yang Liu* Department of Physics, National UniVersity of Singapore, 2 Science DriVe 3, Singapore 117542 ReceiVed: September 15, 2006; In Final Form: October 29, 2006
The effect of long-range attraction on growth models is studied in a two-dimensional colloidal model system. Because of the attraction between the incoming particles and the growing front, incoming monomers are collected preferentially by step protrusions, giving rise to formation of step peaks, the so-called steering effect. However, the situation becomes complicated in the case of incoming dimers. The stronger attraction of the incoming dimers to the existing step particles induces an additional interlayer mass transport, which tends to smoothen out the local step peaks. The long-term effect of the interplay between the steering effect and the smoothing effect is that the local small step peaks are smoothened out and the larger global step protrusions are developed. On the basis of our observations, a mechanism is suggested to interpret the reentrant smooth growth occurring at low temperatures in epitaxial growth.
Introduction Performance of thin films is critically dependent on their surface morphology. Therefore, control of growth processes has been required widely in thin film fabrications. However, to apply a successful control, one needs to determine the appropriate optimal experimental conditions. To do this a complete understanding of the kinetics underlying atomistic processes is an absolute prerequisite. Most previous efforts in exploring the growth kinetics have focused on understanding the role of temperature in mediating various atomistic processes which determine the morphology of the growing surfaces.1-4 In epitexal growth, the appearance of smooth growth at low temperatures is one of the most intriguing phenomena5 and has been studied extensively. Smooth growth or two-dimensional growth at high temperatures is well understood when the interlayer transport is active. At low temperatures when interlayer transport is inhibited by the step-edge barrier, namely, the Ehrlich-Schwoebel (ES) barrier,6,7 three-dimensional growth is presented and the surface becomes rough. However, when the temperature falls below a certain value, smooth growth reappears.5 Several atomistic models were therefore suggested4,8 to address the reentrant smooth growth at low temperatures. Among them, the model of downward funneling (DF) was successful in reproducing the reentrant smooth growth8 and thus accepted and employed widely as an important mechanism in simulations of epitaxial growth.9,10 According to DF mechanism, atoms deposited beyond a step edge would like to funnel down to the lower layers because of their condensation energy. Another interesting finding in epitaxial growth is the existence of the step-adatom attraction.11 Since its first report, the effect of the step-adatom attraction on epitaxial growth has received great attention. It was found that in the case of grazing incidence, the trajectories of the incoming atoms were modified considerably by the local morphology due to the attraction between the growing front and the incoming atom:12 incoming atoms are preferentially adsorbed on the top of the exiting island, the socalled steering effect. Further study revealed that even at normal * To whom correspondence should be addressed. Phone:(65)-65162812. Fax:(65)-6777-6126. E-mail:
[email protected].
incidence, the steering effect can still play a significant role in modifying surface morphology.13 Later, the trajectories of the incoming adatoms beyond the step edges were studied in detail in the presence of an attraction.14 The result showed that the incoming atoms beyond the step edges are preferentially attracted to the uppermost layers rather than funnel down to the lower layers as suggested by DF. Consequently, the presence of the attraction between the incoming atoms and the growing front tends to undermine the DF mechanism. However, DF has been suggested to play a key role in causing the reentrant smooth growth at low temperatures and wildly used in simulations.8-10 When DF fails to work in the presence of the attraction, interpretation of the reentrant smooth growth is again open to question. Most previous works studying the steering effect and DF were based on computer simulations in which experimental conditions have been considerably simplified and the interplay between atomistic processes has been normally neglected. However, in real experiments various atomistic processes proceed at the same time and should influence each other. Generally, the presence of a specific growth model is the outcome of the interplay between different atomistic processes. For example, when incoming atoms approach step edges, there should always be some adsorbed adatoms diffusing on the terraces or along the step edges because incorporation of the adsorbed adatoms into the kink sites is not immediate in practice. Given that the attraction can modify the trajectories of the incoming atoms, how will the attraction from the incoming atoms affect the behavior of the adatoms staying near the step edges? In this report we found from our experimental observations that the effect of the attraction between the incoming atoms and the step atoms actually induces the downward transport as well as the steering effect. Direct investigation of the effect of the attraction on growth models is difficult in atomic systems. Therefore, our study is carried out in a colloidal model system in which a long-range attraction is present. It has been well known that colloidal particles in solutions undergo typical Brownian motion and exhibit collective behaviors analogous to that of typical atomic systems.15,16 For that reason, colloidal suspensions have been
10.1021/jp0660381 CCC: $37.00 © 2007 American Chemical Society Published on Web 12/22/2006
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J. Phys. Chem. C, Vol. 111, No. 3, 2007 1343
Figure 1. Experimental setup. A colloidal suspension is sealed between two pieces of ITO-coated conducting glass plates separated by insulating spacers (the gap between the two glass plates is H ) 120 ( 5 µm). A growing two-dimensional colloidal crystal is presented. The diameter of colloidal particles is 0.99 µm.
extensively used in the past two decades as model systems to address fundamental questions in condensed matter physics, such as melting17,18 and glass transitions.19,20 The kinetics of crystal nucleation has also been studied in colloidal model systems.21,22 These studies have produced a great deal of insight and shed new light on our understanding of long-standing fundamental questions. The objective of this report is to address the kinetics of crystal growth. More specifically, we focused our attention on the effect of the long-range attraction on the growth model. Experimental Methods Figure 1 shows our experimental setup. Monodisperse colloidal particles (polystyrene spheres of diameter 0.99 µm, polydispersity < 5%, Bangs Laboratories) are dispersed uniformly in deionized water. This system has been used to study nucleation21 and two-dimensional melting.23 In our case, a low volume fraction of 0.03% is chosen and the surface potential of the colloidal spheres is adjusted to -72 mV by Na2SO4 (10-4 M). The pH of the suspension is measured at 6.35. The colloidal suspension is then sealed between two parallel horizontal conducting glass plates coated with indium tin oxide (ITO). Once an alternating electric field (AEF) is applied, the fluid flow induced by the AEF transports colloidal particles to the surface of the glass plates where under certain conditions twodimensional crystals are formed. The growth processes are recorded for analysis by a digital camera (CoolSNAP cf, Photometrics) which is mounted on the Olympus BX51 microscope. The result presented in this paper is obtained under the conditions f ) 2000 Hz, Vpp ) 2.5 V (the gap between two glass plates is H ) 120 ( 5 µm). In this system two competing forces act between colloidal particles: the long-range attraction induced by the electrohydrodynamic (EHD) mechanism and the electrostatic repulsion. The EHD mechanism suggests that the presence of a charged particle near the electrode surface distorts the local electric field and local fluid flows are generated around the colloidal particle.
Figure 2. Steering effect resulting from the long-range attraction. (a) Step peaks (pyramidal protrusions) on the growing front. (b, c) Incorporation process of particles A. (d) Trajectory of particle A. (e) Two-dimensional diffusion coefficient of colloidal particles on the glass substrate.
These fluid flows bring colloidal particles toward each other,24,25 and inside the colloidal clusters, the attraction is balanced by the electrostatic repulsion. Results and Discussion The growing two-dimensional crystals in our experiments are characterized by step peaks as shown in Figure 2a. Colloidal particles in this system are homogeneously transported to the growing front. Therefore, formation of the step peaks indicates that there exists a mechanism which causes incoming particles to incorporate preferentially at the positions where step peaks are created. To identify the underlying mechanism of formation of the step peaks, hundreds of incorporation processes of incoming particles are investigated individually. A typical incorporation process occurring at the top of a step peak is presented by snapshots, Figure 2b and c. In this process an incoming particle A diffuses from the solution (Figure 2b) to the peak and eventually becomes incorporated at the top of the peak (Figure 2c). The trajectory of particle A is recorded and shown in Figure 2d. Figure 2d shows that particle A experiences two distinct motions in succession while approaching the step peak: first, it undergoes a Brownian motion; the Brownian motion comes to an end when particle A is about 5 µm away from the step peak and followed by a motion directed along a straight line to the top of the step peak. We find that particle A undergoing Brownian motion shows no tendency to move toward the step peak. It is the following well-directed straight motion which transports particle A to the top of the
1344 J. Phys. Chem. C, Vol. 111, No. 3, 2007 step peak. The motion along a well-directed straight line indicates that there should exist an attraction between particle A and the growing font. The distance traveled along the straight line can serve as a good estimate of the working range of the attractive force. In our experiments the average length of the straight trajectory is measured to be 5-6 µm, which, in comparison with the diameter, 1 µm, of the colloidal particle, is a long-range attraction. To be sure that the behavior of particle A is not caused by the interaction between the colloidal particle and the glass substrate, the two-dimensional diffusion coefficient D2D of colloidal particles far away from the growing front is obtained by measuring the mean square displacement of a diffusing particle as a linear function of investigation interval ∆t. D2D is calculated from the slope of the linear fit of ≈ ∆t by D2D ) /4∆t ) 0.39 ( 0.02 µm2/s. According to the Stokes-Einstein equation, the particle diffusion coefficient can also be calculated by D ) kT/6πηR. In our case, T is 293 K and the corresponding water viscosity η(T) is given by 10-3 Ns m-2. The particle radius is 0.5 µm. Thus, the theoretical diffusion coefficient is given by 0.43 µm2/s, which is in good agreement with our experimental measurement. Therefore, we can conclude that our experimental measurement is believable within the available margin of uncertainty and the ITO glass surface has little effect on the behavior of the colloidal particles. Further confirmation of the existence of the attraction comes from the acceleration of the incoming particles when they approach the growing front. From the diffusion coefficient D2D we estimate the free diffusion velocity of colloidal particles by V2D ) ∆t)1s ) x4D2D ≈ 1.25 µm/s. The velocities of hundreds of particles just prior to their impingement on the step are measured experimentally to be 3.69 µm/s. By comparing this preimpingement velocity 3.69 µm/s with the free diffusion velocity 1.25 µm/s before impingement, we conclude that incoming particles are strongly accelerated as they approach the growing front. The acceleration of incoming particles serves as strong evidence of the existence of attraction. We have now established that the well-directed straight-line motion of particle A in Figure 2 is caused by the attraction between the incoming particles and the growing front rather than any other factor. The incorporation process of particle A in Figure 2 reveals that because of the attraction, incoming particles will be collected preferentially by the tops of the step peaks; thus, in turn the growth of step peaks is enhanced. This picture is consistent with a previous study of the steering effect.14 We conclude here that it is the growth instability induced by the steering effect which contributes to formation of the step peak as shown Figure 2. However, the steering effect studied in previous simulations12,14 reflects only one aspect of the properties of the attraction. In these simulations, it was assumed that incoming atoms approach the step one by one, that all adsorbed atoms have already been incorporated into the exiting step kink sites and cannot move any more. However, this does not hold true in real experiments. In Figure 3a, an adsorbed monomer A stays on the growing front, not far away from the step edges. One dimer consisting of particles B and C is approaching the front. As the dimer moves close to the step, reaching a distance of about several particle diameters away, the attraction from the dimer begins to work on particle A. Particle A then accelerates toward the right (Figure 3b) and gets down to the lower layer (Figure 3c). This process illustrates that it is possible for the
Zhang and Liu
Figure 3. Descending transport triggered by the attraction from the incoming dimer.
Figure 4. Smoothing effect of the attraction from the incoming dimer. (a-d) Step particles are pulled down by the incoming dimer, resulting in reduction of the local roughness.
attraction from the incoming particles to induce some additional descending transport. The process presented in Figure 3 is the simplest case in our experiments. The most popular and more complicated incorporation processes of the incoming clusters are illustrated by Figure 4. In Figure 4a particles marked by 3-6 form the top of the step peak and an incoming dimer consisting of particles 1 and 2 is approaching. When the incoming dimer is close enough to feel the attraction from the peak, its trajectory is directed toward the step peak (Figure 4b). At the same time, particles 3-6 are also influenced by the attraction from the approaching dimer and begin to slide toward the dimer. The attraction from the incoming dimer eventually pulls particles 4 and 6 down to the lower layers (Figure 4d). The incoming dimer also becomes incorporated into the side of the peak. At the end of this process the peak becomes smoother and the peak top is widened and becomes ready to accept more incoming particles. This attraction-induced smoothing effect has never been suggested in previous studies, but it does occur in our experiments. From the above discussion we find that during the growth the role of the attraction between the incoming particles and the growing front is twofold. It works to produce the steering effect as well as induce a smoothing effect. The steering effect leads to creation of step peaks by attracting incoming particles to the uppermost layers, while the smoothing effect gets part of the adsorbed particles down the lower layers. The attractioninduced smoothing effect works effectively in filling up the gaps between the small step peaks and smoothening the local growing front as Figure 5a-c shows. However, the long-term consequence of the attraction is illustrated by Figure 5d: global largestep protrusions are created when the small step peaks are smoothened out during growth. It is clear that the long-term
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J. Phys. Chem. C, Vol. 111, No. 3, 2007 1345
Figure 6. EHD-induced attraction between the incoming clusters and the step particles will be weakened by fluid flow induced by other step particles. Solid arrows: direction of fluid flow induced by incoming clusters A. Dashed arrows: direction of fluid flow induced by step colloidal particles.
Figure 5. Result of the interplay between the steering effect and the smoothing effect. (a-c) The gap between two neighboring small step peaks is filled up by the descending transport induced by the attraction. (d) The long-term consequence of the attraction is represented by the global step protrusions.
consequence can become obvious only when the growth lasts a relatively long time. In thin film growth, to observe the global step protrusions, the thickness of the film has to be larger than a certain value. In previous studies12-14 the smoothing effect of the attraction was not discovered because the influence of the attraction from incoming adatoms on the adsorbed adatoms was not considered overall and the attraction was considered only as a mechanism for producing growth instability and enhancing mound formation. On the basis of our experimental results we conclude that this kind of understanding is not complete. However, we have to mention that the long-range attraction between colloidal particles by nature is distinct from that acting between atoms. In the case of atoms, the attraction exerted on an atom originates directly from other atoms, while the longrange attraction between colloidal particles here proceeds through fluid flow, and hence, the range of the long-range attraction is determined by the scale of fluid flows. It was found that the scale of the fluid flows around colloidal particles or clusters is frequency dependent.26 Given the scale of fluid flow, incoming particles are preferentially captured by the fluid flow induced by step protrusions. Therefore, the role of fluid flow in bringing colloidal particles together to form crystals and producing the steering effect can actually be represented by an attractive force Fh. In their study, Nadal et al. suggested that Fh can be approximated by the Stoke’s force: Fh ≈ 6πηau(r,ω), where a is the radius of the colloidal particle, η is the viscosity of the solvent, and the fluid velocity u(r,ω) at a distance r from the center of the colloidal particle is frequency dependent. From their calculation, Nadal et al. found that for r . a
〈u(r,ω)〉t ) VaAae-r/l(ω)
(1)
where l(ω) is the frequency-dependent characteristic length of the fluid flow induced by the colloidal particles. Both Va and Aa are frequency-dependent constants. For r . l(ω), u(r,ω)decreases as1/r3. In form, Fh is distinct from the attractions used in simulations based on potentials like the Lennard-Jones (LJ) potential12,14 or the embedded atom method (EAM) potential.13,14 However, Yu et al.14 found by comparing the results of the LJ potential and the EAM potentials that the degree of the steering effect does not strongly depend on the details of the interaction potential. In contrast, increasing the cutoff distance of the LJ interaction potential can enhance the steering effect. This means that the most important parameter in steering effect is the range of the attraction. A typical range of attraction in the LJ potential is around two times the atom diameter. However, the range of Fh in this study is estimated as about 5-6 times the diameter of the colloidal particles. This suggests that the steering effect in our experiments should be stronger than in typical atomic systems. The most interesting phenomenon observed in our experiments is the smoothing effect induced by Fh during the growth of the colloidal crystals. The question is whether this mechanism is applicable in typical atomic systems. In atomic systems, the attractive force exerted by an incoming cluster on a step atom is independent of the existence of other step atoms. However, in our system, the attractive forces Fh of incoming clusters have to be exerted on a step particle through fluid flows as Figure 6 shows. When incoming clusters approach steps, the fluid flow around them will be disturbed and usually weakened by the fluid flow induced by other step atoms. Therefore, the final attractive force exerted by incoming cluster A on step particle B (Figure 6) is normally smaller than that indicated by Fh. At this point, we suggest that direct attractions in atomic systems should be more effective than the attraction caused by the EHD mechanism in inducing the smoothing effect. Here, the smoothing effect of the attraction discovered in our experiments offers a mechanism which may contribute to the reentrant smooth growth observed in epitaxial growth at low temperatures.5 It is obvious that the smoothing effect of the attraction will be improved when step peaks become smaller because the particles adsorbed on the top of the step peaks can be more easily reached by the attraction from the incoming particles and thus be ready to be pulled down to the lower layers. This condition is well satisfied in the epitaxial growth conducted at low temperatures. According to nucleation theory, more and smaller islands will be nucleated at low temperatures on the growing surface because of the reduced mobility of adatoms. The reduced mobility of adatoms also leads them to stay near where they landed. Therefore, the adatoms collected through the steering effect stay near the step edges and are especially
1346 J. Phys. Chem. C, Vol. 111, No. 3, 2007 amenable to being pulled down by the attraction from incoming atoms. At this point, the epitaxial growth proceeding at low temperatures offers a good environment for the attractioninduced smoothing effect to work. Nevertheless, the strength of the attraction from the incoming clusters is also an important factor of the smoothing effect. As we have seen in Figure 2, incoming monomers are not likely to induce an interlayer transport due to their small attraction to the step adatoms. On the other hand, incorporation of incoming dimers is often accompanied by the interlayer transport because of the stronger attraction to the step particles. At this point, we argue that a certain number of dimers or bigger incident clusters may promote the occurrence of the flat thin films at low temperatures. Conclusions We find that the attraction between the incoming particle and the growing front can induce the additional interlayer transport as well as generate the steering effect. This observation suggests that when the DF mechanism is suppressed by the presence of an attraction, the attraction-induced interlayer transport can contribute a different mechanism allowing the adatom to descend, thus leading to a smooth growth. Of course, whether this kind of mechanism is strong enough to replace the role of DF is still open to question. We suggest that in future simulations the influence of the attraction from incoming clusters on the existing adatoms near the step edges should be considered as well as the steering effect. Acknowledgment. We are much indebted to Dr. C. Strom for her valuable suggestion and critical reading of the manuscript. References and Notes (1) Jacobsen, J.; Jacobsen, K. W.; Stoltze, P.; Nørskov, J. K. Phys. ReV. Lett. 1995, 74, 2295.
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